Here we have used that \({{C}\mathrm {art}}\)  has finite products.
The bottom horizontal morphism is induced by the morphism \(c \times d \times \mathbb {R}\rightarrow c \times d\) , which is an element of \(I\) .
Hence, the bottom morphism is a weak equivalence in \({{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\) .
Therefore, it follows from [1]} that \({H}_\infty ^{p/i\, I}\)  is symmetric monoidal.
The exact same proof shows that \({H}_\infty ^{p/i\, \ell I}\)  is symmetric monoidal as well.
The fact that \({H}_\infty ^{p/i\, I \ell }\)  is symmetric monoidal will be seen in Corollary REF .
